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Theorem resco 5013
Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)
Assertion
Ref Expression
resco  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )

Proof of Theorem resco
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4817 . 2  |-  Rel  (
( A  o.  B
)  |`  C )
2 relco 5007 . 2  |-  Rel  ( A  o.  ( B  |`  C ) )
3 vex 2663 . . . . . 6  |-  x  e. 
_V
4 vex 2663 . . . . . 6  |-  y  e. 
_V
53, 4brco 4680 . . . . 5  |-  ( x ( A  o.  B
) y  <->  E. z
( x B z  /\  z A y ) )
65anbi1i 453 . . . 4  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  ( E. z ( x B z  /\  z A y )  /\  x  e.  C )
)
7 19.41v 1858 . . . 4  |-  ( E. z ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( E. z ( x B z  /\  z A y )  /\  x  e.  C )
)
8 an32 536 . . . . . 6  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
9 vex 2663 . . . . . . . 8  |-  z  e. 
_V
109brres 4795 . . . . . . 7  |-  ( x ( B  |`  C ) z  <->  ( x B z  /\  x  e.  C ) )
1110anbi1i 453 . . . . . 6  |-  ( ( x ( B  |`  C ) z  /\  z A y )  <->  ( (
x B z  /\  x  e.  C )  /\  z A y ) )
128, 11bitr4i 186 . . . . 5  |-  ( ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  ( x
( B  |`  C ) z  /\  z A y ) )
1312exbii 1569 . . . 4  |-  ( E. z ( ( x B z  /\  z A y )  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
146, 7, 133bitr2i 207 . . 3  |-  ( ( x ( A  o.  B ) y  /\  x  e.  C )  <->  E. z ( x ( B  |`  C )
z  /\  z A
y ) )
154brres 4795 . . 3  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  ( x ( A  o.  B ) y  /\  x  e.  C ) )
163, 4brco 4680 . . 3  |-  ( x ( A  o.  ( B  |`  C ) ) y  <->  E. z ( x ( B  |`  C ) z  /\  z A y ) )
1714, 15, 163bitr4i 211 . 2  |-  ( x ( ( A  o.  B )  |`  C ) y  <->  x ( A  o.  ( B  |`  C ) ) y )
181, 2, 17eqbrriv 4604 1  |-  ( ( A  o.  B )  |`  C )  =  ( A  o.  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316   E.wex 1453    e. wcel 1465   class class class wbr 3899    |` cres 4511    o. ccom 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-co 4518  df-res 4521
This theorem is referenced by:  cocnvcnv2  5020  coires1  5026  relcoi1  5040  dftpos2  6126
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